Given a measurable space ($\Omega$,$\mathcal{F}$) and two probability measures $\mu$, $\nu$. We define total variation distance as
$$\lVert \mu - \nu \rVert_{TV} = \sup_{A\in \mathcal{F}}|\mu(A)-\nu(A)|.$$
Does it always exist $A\in\mathcal{F}$ such that $\lVert \mu - \nu \rVert_{TV}=|\mu(A)-\nu(A)|$?
If the answer is no, is there a condition such that it is true?
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